The degree of \(x\in X\)  in the family \(\mathcal {F}\)  is the number of sets that contain \(x\) .
Over 30 years ago Beck and Fiala [1]} proved that
if the maximum degree of vertices in \(\mathcal {F}\)  is \(d\) , then \(\operatorname{disc}\mathcal {F}\le 2d-2\) .
The remarkable feature of the result is that it depends neither on the number of sets in \(\mathcal {F}\) 
nor on the size of \(X\) . If dependence on these quantities is permitted, the best result
is due to Banaszczyk [2]}, and asserts that \(\operatorname{disc}\mathcal {F}\le c\sqrt{d \log \vert X\vert }\) .
Let \(f(d)\stackrel{\text{\tiny {def}}}{=}\max \operatorname{disc}\mathcal {F}\) , where the maximum is taken over all set families of degree at most \(d\) .
In [1]} Beck and Fiala conjectured that \(f(d)=O(\sqrt{d})\) . If true,
the conjecture would be a strengthening of Spencer's six deviation theorem [4]}.
A related, but stronger conjecture was made by János Komlós [4]}.
A relaxation of Komlós's conjecture to vector-valued colorings has been established by Nikolov [6]}.
For a general overview of discrepancy theory see books [7]}, [8]}.
